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dDsC dispersion correction

The expression for dispersion energy within the dDsC dispersion correction [139,138] (DFT-dDsC) is very similar to that of DFT-D2 method (see eq. 6.87), the important difference is, however, that the dispersion coefficients and damping function are charge-density dependent. The dDsC method is therefore able to take into account variations in vdW contributions of atoms due to their local chemical environment. In this method, polarizability, dispersion coeficients, charge and charge-overlap of an atom in molecule or solid are computed in the basis of a simplified exchange-hole dipole moment formalism,[139] pioneered by Becke and Johnson[140].

The dDsC dispersion energy is expressed as follows

$\displaystyle {{E}_{\mathrm{disp}}}=-\sum\limits_{i=2}^{{{N}_{\mathrm{at}}}}{\sum\limits_{j=1}^{i-1} {{{f}_{6}}(b{{R}_{ij}})\frac{C_{6,ij}}{R_{ij}^{6}}}}$ (6.110)

where $ N_{\mathrm{at}}$ is the number of atoms in the system and $ b$ is the Tang and Toennies (TT) damping factor. The damping function $ f_{6}(bR_{ij})$ is defined as follows

$\displaystyle f_{6}(x)=1-{\rm exp}(-x)\sum^{6}_{k=0}\frac{x^k}{k!}$ (6.111)

and its role is to attenuate the correction at short internuclear distances. A key component of the dDsC method is the damping factor $ b$:

$\displaystyle b(x)=\frac{2 b_{ij,\mathrm{asym}}}{{{e}^{{{a}_{0}}\cdot x}}+1} ,$ (6.112)

where the fitted parameter $ a_0$ controls the short-range behavior and $ x$ is the damping argument for the TT-damping factor associated with two separated atoms ( $ b_{ij,\mathrm{asym}}$). The term $ b_{ij,\mathrm{asym}}$ is computed according to the combination rule:

$\displaystyle b_{ij,\mathrm{asym}}=2\frac{b_{ii,\mathrm{asym}}\cdot b_{jj,\mathrm{asym}}}{b_{ii,\mathrm{asym}} + b_{jj,\mathrm{asym}}}$ (6.113)

with $ b_{ii,\mathrm{asym}}$ being estimated from effective atomic polarizabilities:

$\displaystyle {{b}_{ii,\mathrm{asym}}}={{b}_{0}}\cdot \sqrt[3]{\frac{1}{{{\alpha }_{i}}}}$ (6.114)

The effective atom-in-molecule polarizabilities $ {\alpha }_{i}$ are computed from the tabulated free-atomic polarizabilities (available for the elements of the first six rows of the periodic table except of lanthanides) in the same way as in the method of Tkatchenko and Scheffler (see Sec. 6.77.3) but the Hirshfeld-dominant instead of the conventional Hirshfeld partitioning is used. The last element of the correction is the damping argument $ x$

$\displaystyle x=\left( 2{{q}_{ij}}+\frac{\vert({{Z}_{i}}-N_{i}^{D})\cdot ({{Z}_...
...)\vert}{{{r}_{ij}}} \right)\frac{N_{i}^{D}+N_{j}^{D}}{N_{i}^{D}\cdot N_{j}^{D}}$ (6.115)

where $ Z_i$ and $ N_i^D$ are the nuclear charge and Hirshfeld dominant population of atom $ i$, respectively. The term $ 2q_{ij} = q_{ij} + q_{ji}$ is a covalent bond index based on the overlap of conventional Hirshfeld populations $ q_{ij}=\int w_i({\mathbf{r}})w_j({\mathbf{r}})\rho({\mathbf{r}})d{\mathbf{r}}$, and the fractional term in the parentheses is a distance-dependent ionic bond index.

The DFT-dDsC calculation is invoked by setting IVDW=4. The default values for damping function parameters are available for the functionals PBE (GGA=PE) and revPBE (GGA=RP). If other functional is used, the user must define these parameters via corresponding tags in INCAR (parameters for common DFT functionals can be found in Ref. [138]) The following parameters can be optionally defined in INCAR:
VDW_RADIUS = 50.0 cutoff radius (Å) for pair interactions
VDW_S6 = 13.96 scaling factor $ {a}_{0}$
VDW_SR = 1.32 scaling factor $ {b}_{0}$

Performance of PBE-dDsC in description of the adsorption of hydrocarbons on Pt(111) has been examined in Ref. [141] PCCP 17, 28921 (2015).


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